Ultrasound waveform tomography with tv regularization

ABSTRACT

Synthetic-aperture ultrasound tomography systems and methods using scanning arrays and algorithms configured to simultaneously acquire ultrasound transmission and reflection data, and process the data for improved ultrasound tomography imaging, wherein the tomography imaging comprises total-variation regularization, or a modified total variation regularization.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a 35 U.S.C. §111(a) continuation of PCT international application number PCT/US2013/024545 filed on Feb. 3, 2013, incorporated herein by reference in its entirety, which claims priority to, and the benefit of, U.S. provisional patent application Ser. No. 61/594,865, filed on Feb. 3, 2012, incorporated herein by reference in its entirety. Priority is claimed to each of the foregoing applications.

The above-referenced PCT international application was published as PCT International Publication No. WO 2013/116809 on Aug. 8, 2013, incorporated herein by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under Contract No. DE-AC52-06NA25396 awarded by the Department of Energy. The Government has certain rights in the invention.

INCORPORATION-BY-REFERENCE OF COMPUTER PROGRAM APPENDIX

Not Applicable

NOTICE OF MATERIAL SUBJECT TO COPYRIGHT PROTECTION

A portion of the material in this patent document is subject to copyright protection under the copyright laws of the United States and of other countries. The owner of the copyright rights has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the United States Patent and Trademark Office publicly available file or records, but otherwise reserves all copyright rights whatsoever. The copyright owner does not hereby waive any of its rights to have this patent document maintained in secrecy, including without limitation its rights pursuant to 37 C.F.R. §1.14.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention pertains generally to imaging, and more particularly to ultrasound imaging using a synthetic aperture ultrasound ray tomography and ultrasound waveform tomography.

2. Description of Related Art

Breast cancer is the second-leading cause of cancer death among American women. The breast cancer mortality rate in the U.S. has been flat for many decades, and has decreased only about 20% since the 1990s. Early detection is the key to reducing breast cancer mortality. There is an urgent need to improve the efficacy of breast cancer screening. Ultrasound tomography is a promising, quantitative imaging modality for early detection and diagnosis of breast tumors.

Ultrasound waveform tomography is gaining popularity, but is computationally expensive, even for today's fastest computers. The computational cost increases linearly with the number of transmitting sources.

Synthetic-aperture ultrasound has great potential to significantly improve medical ultrasound imaging. In a synthetic aperture ultrasound system, ultrasound from each element of a transducer array propagates to the entire imaging domain, and all elements in the transducer array receive scattered signals.

Many conventional ultrasound systems record only 180° backscattered signals. Others are configured to receive only transmission data from the scanning arrays. Accordingly, these systems suffer from extensive computational costs, insufficient resolution, or both.

Waveform inversion can be implemented either in the time domain, or in the frequency domain. Because of the ill-posedness caused by the limited data coverage, multiple local-minimum solutions exist, and therefore, certain stabilization numerical techniques need to be incorporated within inversion process to obtain a global-minimum solution. In recent years, many approaches have been developed for this purpose. Regularization techniques can be employed to alleviate the instability of the original problem. Preconditioning approaches can also be used in waveform inversion to create a well-conditioned problem with lower dimensions. In addition, prior information about the model can be introduced to improve the convergence of waveform inversion.

BRIEF SUMMARY OF THE INVENTION

The system and method of the present invention uses ultrasound data acquired using a synthetic-aperture ultrasound system. The investigational synthetic-aperture ultrasound tomography system of the present invention allows acquisition of each tomographic slice of patient ultrasound data in real time. In the system, each element of the transducer array transmits ultrasound sequentially, and elements in the transducer array simultaneously record ultrasound signals scattered from the tissue after each element is fired. The features of the system and method of the present invention provide a real-time synthetic-aperture system that can be used for patient data acquisition.

In the synthetic-aperture ultrasound tomography system of the present invention, ultrasound from each element of a transducer array or a virtual source of multiple elements propagates to the entire imaging domain, and all elements in the transducer array receive ultrasound signals reflected/scattered from the imaging region and/or transmitted/scattered through the imaging region. Therefore, the acquired synthetic-aperture ultrasound data contain information of ultrasound reflected/scattered and transmitted from all possible directions from the imaging domain to the transducer array to generate a more accurate, 3-D, high resolution image, while minimizing computational costs of the system.

One aspect of the invention is an ultrasound waveform tomography method with the total-variation (TV) regularization to improve sound-speed reconstructions of small breast tumors. The nonlinear conjugate gradient (NCG) method is used to solve waveform inversion with the TV regularization. The gradient of the misfit function is obtained using an adjoint state method.

Another aspect of the invention is a novel ultrasound waveform tomography method with a modified total-variation regularization scheme, in which a separate regularization term is added, such that the edge-preserving can be more effective without adding too much extra computational cost.

We minimize the misfit function using an alternating minimization algorithm. The cost function is decomposed with the modified TV regularization into two regularization problems, a L₂-norm-based Tikhonov regularization problem and a L₁-norm-based TV regularization problem. The nonlinear conjugate gradient (NCG) approach is used to solve for the first Tikhonov regularization problem. Then, an adjoint state method is used to compute the gradient of the misfit function. The split-Bregman method to is used to solve the second regularization problems. In one embodiment, the use of the split-Bregman method allows for computations that are (a) it is computationally efficient; and (b) the selection of the smoothing parameter in the TV regularization term can be avoided.

Further aspects of the invention will be brought out in the following portions of the specification, wherein the detailed description is for the purpose of fully disclosing preferred embodiments of the invention without placing limitations thereon.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

The invention will be more fully understood by reference to the following drawings which are for illustrative purposes only:

FIG. 1 is a schematic diagram of a synthetic-aperture ultrasound system in accordance with the present invention.

FIG. 2 is a schematic diagram of a synthetic-aperture ultrasound tomography system for scanning breast tissue in accordance with the present invention

FIG. 3 is a schematic diagram of the scanner of the ultrasound tomography system of FIG. 1 interrogating a region of tissue.

FIG. 4 shows flow diagram of a method for sequentially exciting a region of tissue and acquiring reflection and transmission data in accordance with the present invention.

FIG. 5 illustrates a schematic view of a two parallel-bar ultrasound transducer array scanner.

FIG. 6 illustrates a schematic view of a scanner comprising two parallel planar arrays.

FIG. 7 shows a schematic view of a cylindrical array scanner having a cylindral 2-D array of transducers and a 2-D planner array at the bottom of the cylinder.

FIG. 8 shows a flat transducer configured to generate a collimated beam.

FIG. 9 shows an arcuate transducer configured to generate a diverging beam.

FIG. 10 shows a schematic view of a torroidal array scanner having a a circular array of transducers.

FIG. 11 shows a schematic view of a synthetic-aperture ultrasound breast tomography scanner that incorporates use of two circular transducer arrays.

FIG. 12 shows a schematic view of a scanner comprising a semicircular or arcuate array having transducers in an opposing or facing orientation with planar array.

FIG. 13 illustrates a scanner that reduces the 2D arrays in FIG. 12 to 1D arrays.

FIG. 14 is a flow diagram of a synthetic aperture ultrasound tomography method in accordance with the present invention.

FIG. 15 shows an image of a numerical breast phantom containing two different tumors.

FIG. 16A and FIG. 16B show imaging results (tomographic reconstruction in FIG. 16A, and vertical profile along the center of the tumors in FIG. 16B) obtained using only the reflection data.

FIG. 17A and FIG. 17B show imaging results (tomographic reconstruction in FIG. 17A, and vertical profile along the center of the tumors in FIG. 17B) obtained using only the transmission data.

FIG. 18A and FIG. 18B show imaging results (tomographic reconstruction in FIG. 18A, and vertical profile along the center of the tumors in FIG. 18B) obtained using both transmission and reflection data simultaneously in accordance with method of the present invention.

FIG. 19 illustrates a method for generating the ultrasound waveform step of FIG. 14 using both transmission and reflection data for ultrasound waveform tomography.

FIG. 20A through FIG. 20B show tomographic reconstruction results for tomographic reconstruction with a modified total-variation regularization scheme using only reflection data.

FIG. 21A through FIG. 21B show results for tomographic reconstruction with a modified total-variation regularization scheme obtained using only transmission data.

FIG. 22A through FIG. 22B show results for tomographic reconstruction with a modified total-variation regularization scheme obtained using both transmission and reflection data in accordance with the present invention.

FIG. 23 shows a flow diagram for a computational method for solving the TV regularization problem of the present invention.

FIG. 24 shows a flow diagram for a computational method for solving the modified TV regularization problem of the present invention.

FIG. 25 shows an algorithm using NCG to solve a step in the method of FIG. 24.

FIG. 26 shows an algorithm using Split Bregman iteration to solve a step in the method of FIG. 24.

FIG. 27A and FIG. 27B are images showing a numerical breast phantom containing two small breast tumors.

FIG. 28A through FIG. 28C show sound-speed reconstructions obtained using ultrasound waveform tomography with (FIG. 28A) Tikhonov regularization, (FIG. 28B) TV regularization, and (FIG. 28C) modified TV regularization.

FIG. 29A through FIG. 29C show horizontal profiles of the waveform tomography reconstructions through the center of the large tumor with (FIG. 29A) Tikhonov regularization, (FIG. 29B) TV regularization, and (FIG. 29C) modified TV regularization.

FIG. 30A through FIG. 30C show horizontal profiles of the waveform tomography reconstructions through the center of a small tumor with (FIG. 30A) Tikhonov regularization, (FIG. 30B) TV regularization, and (FIG. 30C) modified TV regularization.

FIG. 31A through FIG. 31C show images of numerical breast phantom FIG. 31A containing two tumors with sizes of 0.75 mm (half a wavelength) and 1.5 mm (1 wavelength), TV reconstruction (FIG. 31B), along with modified TV reconstruction (FIG. 31C).

FIG. 32A and FIG. 32B are images showing horizontal profiles of the result of ultrasound waveform tomography with TV regularization through the center of the large tumor (FIG. 32A), and through the center of the small tumor (FIG. 32B).

FIG. 32C and FIG. 32D are images showing horizontal profiles of the result of ultrasound waveform tomography with modified TV regularization through the center of the large tumor (FIG. 32C), and through the center of the small tumor (FIG. 32D).

FIG. 33A through FIG. 33C show images of numerical breast phantom (FIG. 33A) containing two tumors with sizes of 1.5 mm (1 wavelength) and 2.25 mm (1.5 wavelengths), TV reconstruction (FIG. 33B), along with modified TV reconstruction (FIG. 33C).

FIG. 34A and FIG. 34B are images showing horizontal profiles of the result of ultrasound waveform tomography with TV regularization through the center of the large tumor (FIG. 34A), and through the center of the small tumor (FIG. 34B).

FIG. 34C and FIG. 34D are images showing horizontal profiles of the result of ultrasound waveform tomography with modified TV regularization through the center of the large tumor (FIG. 34C), and through the center of the small tumor (FIG. 34D).

FIG. 35A through FIG. 35C show images of numerical breast phantom FIG. 35A containing two tumors with sizes of 3.0 mm (2 wavelengths) and 3.75 mm (2.5 wavelengths), TV reconstruction (FIG. 35B), along with modified TV reconstruction (FIG. 35C).

FIG. 36A and FIG. 36B are images showing horizontal profiles of the result of ultrasound waveform tomography with TV regularization through the center of the large tumor (FIG. 36A), and through the center of the small tumor (FIG. 36B).

FIG. 36C and FIG. 36D are images showing horizontal profiles of the result of ultrasound waveform tomography with modified TV regularization through the center of the large tumor (FIG. 36C), and through the center of the small tumor (FIG. 36D).

FIG. 37A through FIG. 37C show images of numerical breast phantom FIG. 37A containing two tumors with sizes of 3.75 mm (2.5 wavelengths) and 4.5 mm (3 wavelengths), TV reconstruction (FIG. 37B), along with modified TV reconstruction (FIG. 37C).

FIG. 38A and FIG. 38B are images showing horizontal profiles of the result of ultrasound waveform tomography with TV regularization through the center of the large tumor (FIG. 38A), and through the center of the small tumor (FIG. 38B).

FIG. 38C and FIG. 38D are images showing horizontal profiles of the result of ultrasound waveform tomography with modified TV regularization through the center of the large tumor (FIG. 38C), and through the center of the small tumor (FIG. 38D).

DETAILED DESCRIPTION OF THE INVENTION

The description below is directed to synthetic aperture ultrasound tomography systems for imaging a medium such as patient tissue, along with ultrasound waveform tomography methods for acquiring and processing data acquired from these systems, or other systems that may or may not be available in the art.

The synthetic-aperture breast ultrasound tomography system of the present invention uses synthetic-aperture ultrasound to obtain quantitative values of mechanical properties of breast tissues. In this system, each transducer element transmits ultrasound waves sequentially, and when an ultrasound transducer element transmits ultrasound waves propagating through the breast, all ultrasound transducer elements (at least within a portion of an array) simultaneously receive ultrasound reflection/transmission, or forward and backward scattering signals. The ultrasound reflection/transmission signals are used to obtain quantitative values of mechanical properties of tissue features (and in particular breast tumors), including the sound speed, density, and attenuation.

While the systems and methods described below are particularly directed and illustrated for imaging of breast tissues, it is appreciated that the systems and methods may also be employed for waveform tomography on other tissues or scanning mediums.

I. Synthetic Aperture Ultrasound Tomography System

FIG. 1 is a schematic diagram of a synthetic-aperture ultrasound system 10 in accordance with the present invention. The system 10 includes a scanner 12 comprising a plurality of individual transducer elements 16 disposed within one or more arrays (e.g. the opposing parallel arrays 14 a and 14 b shown in FIG. 1). The scanner 12 is coupled to a server or like computing apparatus 20 (e.g. with a cable 15 or other connection means such as, but not limited to, a wireless connections means) and synthetic aperture ultrasound data acquisition system 18 that outputs RF data 28 corresponding to readings acquired by the scanner 12.

The computer 20 comprises a processor 24 configured to operate one or more application programs 22 located within memory 25, wherein the application programs 22 may contain one or more algorithms or methods of the present invention for imaging a tissue medium for display via a graphical user interface 23 on monitor 26, or other means. For example, the application programming 22 may comprise the programming configured for operating the sequential excitation method 50 shown in FIG. 4 or ultrasound waveform tomography imaging method 200 shown in FIG. 14. The computer 20 controls ultrasound tomography data acquisition, and the process is completed automatically. The whole-breast scanning time with approximately 100 slides takes approximately 2 minutes.

FIG. 2 is a schematic view of a breast ultrasound tomography system 11 in accordance with the present invention. System 11 includes a table 70 having a water tank 76 with an open aperture at the top of the table 70 for insertion of the patient's breast tissue (which ideally hangs pendant within water tank 76 during imaging). Tank 76 includes one or more synthetic-aperture ultrasound transducer arrays 74 located within one or more surfaces of the tank. The transducer array(s) 74 are immersed within the water tank 76 configured for receiving the patients breast 44 through aperture 72, and scanning the breast 44 while the patient is lying down on the table 70 in the prone position. As described in further detail below, transducer array(s) 74 may comprise a number of different configurations, with the water tank housing 76 shaped accordingly to house the array(s) 74. The water tank housing 76 material preferably comprises a light, non-conductive material that conforms to the shape of the array(s) 74 (e.g. rectangular for 2-parallel bar array scanner 12 of FIG. 1, or cylindrical for the scanners 110, 120 and 130 shown in FIG. 7, FIG. 10 and FIG. 11, respectively).

Positioning of the active areas of all array(s) 74 relative to the water tank housing 76 is preferrably aligned such that the ultrasound energy for the transducer elements 16 (FIG. 1) is focused onto the same plane perpendicular to the housing (for parallel bar scanner 12 (FIG. 5) or planar 100 (FIG. 6) arrays). The arrays (e.g. arrays 14 a and 14 b, FIG. 1) are preferrably electrically isolated and grounded.

The system 11 includes a data acquisition system 18 that may be coupled to a computer system or electronics 78 that control scanning. The data acquisition system 18 may also be coupled to a computer 20 for running application programming 22 (FIG. 1) to perform tomography reconstructions.

During the ultrasound data acquisition in the synthetic-aperture ultrasound tomography system 10, the raw ultrasound data 28 (radio-frequency data) may be first stored within computer memory 25 (FIG. 1) (which may comprise solid state drives or other storage means available in the art), allowing real-time patient data acquisition for clinical applications.

FIG. 3 is a schematic diagram of the two parallel bar arrays 14 a and 14 b of scanner 12 of FIG. 1 shown interrogating a region of tissue 44 (e.g. breast tissue for mammography) in accordance with a preferred method of the present invention. The ultrasound imaging system 10 focuses an array 14 a and 14 b of N transducers 16 acting in a transmit-receive mode. Each element of the array 14 a 14 b is excited sequentially (e.g. transducer 3 of array 14 a is shown in excitation mode) to generate an ultrasound field or signal 30 through the tissue surface 40 and into tissue medium 44 having a plurality of point scatterers 42. The backscattered signals 32 are measured in parallel by all N elements 16. In addition, opposing array 14 b transducers are positioned facing array 14 a such that one or more elements of the array 14 b receive direct transmission signals 30 simultaneously with reception of backscatter or reflection signals 32 being received by array 14 a.

FIG. 4 shows flow diagram of a method 50 for sequentially exciting a region of tissue 44 in accordance with the present invention. At step 52, a first element (e.g. element 1 or i) of array 14 a 14 b of N ultrasound transducer elements 16 is excited for interrogating an inhomogeneous medium 44. At step 54, the backscattered/reflected signals 32 are received/measured by all elements 16 (of at least 14 a), while transmission signals 30 are received/measured by one or more elements 16 of array 14 b. At step 58, the method evaluates whether all the elements 16 in the arrays 14 a and 14 b have been excited (and imaged). If the last element in the arrays 14 a, 14 b has not been reached, the method moves to the next element 16 in the array (14 a or 14 b) at step 60, and repeats the process sequentially until the N^(th) element is reached. At this point, the individual reflection/transmission data are RF data, and the process 50 transfers the RF data to memory or solid state drives 25 at step 64.

In the phased transducer arrays for synthetic-aperture breast ultrasound tomography, a plurality of transducer elements 16 are fired with different delayed times to simulate ultrasound waves emerging from a virtual point source. The systems and methods of the present invention preferrably use the virtual point sources of the synthetic-aperture breast ultrasound tomography system to improve signal-to-noise ratios of breast ultrasound data.

The various scanning arrays invention, described below with reference to FIG. 5 through FIG. 7 and FIG. 10 through FIG. 13, are shown to illustrate that the systems 10, 11 and methods 50, 200 may be achieved in various configurations. Yet, the scanning arrays of FIG. 5 through FIG. 7 and FIG. 10 through FIG. 13 all share at least one common characteristic in that at a plurality of transducers 16 of an array, or portion of an array, oppose (at a spaced-apart distance across the target scanning medium 44) a plurality of transducers 16 of either another portion of the array, or a separate array, so that reflection and transmission data may be acquired with each successive transducer excitation. The following are specific examples of arrays that may be used in the systems 10, 11 and methods 50, 200 of the present invention. However, other configurations are contemplated. In each of these configurations, the scanner 74 is shown without table 70 or housing 76 for clarity.

A. Dual Parallel-Bar Array Scanner

FIG. 5 illustrates a two parallel-bar ultrasound transducer array scanner 12, which is illustrated in reference to implementation within system 10 in FIG. 1, and schematically in operation as a synthetic-aperture scanner in FIG. 3.

As shown in FIG. 5, the two arrays 14 a and 14 b are shown in opposing orientation (e.g facing eachother and matching in location along x-axis in FIG. 5), and positioned in the x-y plane (preferrably parallel to table 70 in FIG. 2, such that they are spaced-apart across the scanning region 44. Each of the 14 a and 14 b comprises a plurality of N transducers 16 (e.g. count of 128) linearly aligned in series (shown in along the x-axis for reference) as parallel-phased arrays firing toward each other in operation (see FIG. 3).

A robotic stage 90 is provided so that the arrays can move in unison vertically along the z-axis to scan the tissue 44. The transducer arrays 14 a and 14 b are configured to scan the breast 44 from the chest wall to the nipple region, slice by slice. To image the axillary region (region of breast closest to the armpit of the patient, not shown), the two transducer arrays 14 a and 14 b can be steered toward the axillary region, with one of the transducer arrays placed near the axillary region. The axillary region, or basin, is important to oncologic surgeons, as it represents the principal lymphatic drainage region of the breast. Lymphatic metastasis from a malignant breast lesion will most often occur in this region.

Arrays 14 a and 14 b may also be translated (either in concert, or with respect to each other) in the x and y axes to closely conform to varying patient anatomy.

Referring to FIG. 8 and FIG. 9, the transducer 16 may either be flat or circular, and the surface of the transducer element 16 may either be flat, as in transducer 16 a in FIG. 8, or arcuate in shape, as shown in transducer 16 b of FIG. 9. The flat transducer 16 a of FIG. 8 generates a collimated beam 17, whereas the curvilinear transducer 16 b of FIG. 9 has a focal point P that is behind the emitting surface to generate a diverging beam 19 (defocused or lens configuration preferably in the y-z plane) across a field of view from A to B (centered on C). The curvilinear transducer 16 b of FIG. 9 helps get a 3-D volume while scanning, and is particularly useful with line arrays such as those in FIG. 5, FIG. 10, FIG. 11, and FIG. 13.

In one embodiment, exemplary dimensions for the arrays 14 a and 14 b and transducers 16 are as follows: a length inside the water tank along X-axis (the horizontal direction) of 16 inches, with 19.2 inches along Y-axis (the horizontal direction) and 16 inches in height along Z-axis (the vertical direction). The distances from the ends of the ultrasound phased transducer arrays 14 a and 14 b to the inside walls of the water tank along X-axis are approximately 3.8425 inches. In one embodiment, the horizontal distance between the front surfaces of the two parallel phased ultrasound transducer arrays can be adjusted from 12 cm to 25 cm, with a 1 cm increment utilizing 14 different sets of spacer blocks. The accuracy and precision of the horizontal position is ideally 5 microns or better. The vertical travel (Z axis) of the two parallel ultrasound phased transducer arrays 14 a and 14 b is 10 inches from the top surface of the water level. The vertical travel step interval can be adjusted to any value, such as 0.25 mm, 0.5 mm, 1 mm, and 2 mm.

In one embodiment, array 14 a, 14 b parameters are as follows: center frequency of 1.5 MHz, bandwidth of ˜80% bandwidth (−6 dB) (measured for two-way sound propagation energy), the open angle of ultrasound waves emitting from a single element at ˜80°, with uniform transducer elements 16 (<1 dB variation, and uniform bandwidth for one-way sound propagation energy).

In one embodiment, the arrays 14 a, 14 b comprise 1.5 MHz arrays with 384 elements each, equally spaced along the array. In one example, the dimensions characteristics of the transducer elements are as follows: elevation aperture: 15 mm, element width: 0.4 mm for 1.5 MHz arrays, elevation focus: 10 cm away from the transducer element, with all transducers configured to be aligned along the array and perpendicular to the elevation plane.

It is appreciated that the above dimensions and configuration details are for reference purposes only, and such characteristics may be varied accordingly.

The advantage of the configuration of scanner 12, over, e.g. the planar arrays of FIG. 6, is that the system 10 is using a fewer number of transducer elements.

B. Dual Parallel Planar Array Scanner

FIG. 6 illustrates a scanner 100 comprising two parallel planar arrays 102 a and 102 b aligned opposing each other across the scanning medium 44. Arrays 102 a and 102 b each comprise matching grids of 2-D arrays of transducers 16 (e.g. transducers 16 share the same locations in their respective x-z planes shown in FIG. 6). With the planar arrays the scanner 100 generally does not need to be translated in the z (vertical) direction.

There are generally two limitations for the synthetic-aperture breast ultrasound tomography with the cylindrical or circular transducer arrays: (a) it is difficult to image the axillary region of the tissue 44; and (b) one size of the cylindrical or circular transducer array will either be undersized or oversized for most sizes of the breast.

Synthetic-aperture breast ultrasound tomography with two parallel planar ultrasound transducer arrays 102 a and 102 b can overcome these two limitations. As shown in FIG. 6, one planar/2D transducer array 102 b can be placed close to the axillary region of the tissue 44. In addition, the distance between the two planar ultrasound transducer arrays 102 a and 102 b can be adjusted with respect to each other (either manually or with robotic stage 90 as shown in FIG. 5) to fit different sizes of the breast. The ultrasound transducer elements 16 can be in circular or rectangular shape, and the surface of the transducer element can be either flat or arc-shaped, as shown in FIG. 8 and FIG. 9.

C. Cylindrical Array Scanner

FIG. 7 shows a cylindrical array scanner 110 having a cylindrical 2-D array 112 a of transducers 16 in the inside surface of the cylinder wall 118 of the ultrasound transducer array. A planar array of elements 112 b may also be positioned on the bottom surface 116 of the cylinder, which would primarily capture backscattered signals.

With the singular cylindrical array scanner 110, a first half of the semi-cylinder elements 16 will be opposed to or facing the second half of the semi-cylinder elements 16, and thus be positioned to receive direct transmission signals 30 (see FIG. 3) at least at varying degrees of angles of incidence. Thus depending on the amount of defocusing within each transducer, a plurality, or all, of the non-emitting transducers 16 will be able to receive a direct transmission signal 30 (FIG. 3) (at varying degrees) from the emitting transducer 16, leading to a full 3D ultrasound tomography image of the breast.

The top end 114 of the cylinder is open, such that the breast tissue 44 is immersed into the cylindrical array scanner 110 with 2D ultrasound transducer elements 16 surrounding the tissue 44. As with previous embodiments, the ultrasound transducer elements 16 can be in circular or rectangular shape, and the surface of the transducer element can be either flat or arc-shaped, as shown in FIG. 8 and FIG. 9.

D. Torroidal (Circular) Array Scanner

FIG. 10 shows a torroidal array scanner 120 having a circular array 122 of transducers 16 aligned in a ring that is configured to encircle the breast 44. A robotic stage 124 may be provided to allow for translation of the array 122 to and scan the breast 44 from the chest wall to the nipple region, slice by slice.

With the singular torroidal array scanner 120, a first half of the semi-circle elements 16 will be opposed to or facing the second half of the semi-circle elements 16, and thus be positioned to receive direct transmission signals 30 (see FIG. 3) at least at varying degrees of angles of incidence. Thus, depending on the amount of defocusing within each transducer, a plurality, or all, of the non-emitting transducers 16 will be able to receive a direct transmission signal 30 (at varying degrees) from the emitting transducer 16.

The circular array 122 preferably comprises defocused lens-transducer elements 16 b as shown in FIG. 9, enabling 3-D breast ultrasound tomography. One advantage of the torroidal configuration 120 is using a fewer number of transducer elements compared to the cylindrical transducer array 110.

E. Dual Torroidal (Circular) Array Scanner

FIG. 11. shows another synthetic-aperture ultrasound breast tomography scanner 130 that incorporates use of two circular transducer arrays (upper cirular array 132 a and lower circular array 132 b).

Image resolution depends, at least in part, on ultrasound illumination of the target medium 44. To increase the ultrasound out-of-plane illumination angle, an acoustic diverging lens 16 b, as shown in FIG. 9, may be used to widen the elevation beam to the desired level (e.g. between points B and C in the upper cirular array 132 a and D and E in the lower circular array 132 b (conically diverging beam)). Thus, the defocused ultrasound transducer elements 16 b transmit ultrasound waves propagating not only to the transducer elements within the same circular array, e.g. between B and C in the upper ring 132 a, but also to the other circular transducer array, e.g. between D and E in the lower ring 132 b. The upper transducer array 132 a may be configured to scan the breast 44 from the chest wall position to the nipple region. At each position, the lower transducer array 132 b may move to different vertical position in the z-axis to acquire ultrasound data. This configuration leads to improved vertical resolution of breast ultrasound tomography images compared that obtained using one circular transducer array as shown in FIG. 10.

In practice, the two circular ultrasound transducer arrays 132 a and 132 b are immersed into the water tank 76 and both encircle the breast 44. One or both arrays 132 a and 132 b may be configured to translate vertically via a motorized stage 134. For example, during an ultrasound scan, the upper cirular array 132 a can be positioned against the chest wall, while the lower cirular array 132 b moves upward from below the nipple region, or vice versa.

As with previous embodiments, each element of one transducer array is fired sequentially, and all elements of both transducer arrays receive ultrasound scattering data 32. The scanner 130 acquires not only ultrasound propagating from one element to all elements within the same transducer array, but also those ultrasound waves propagating from the emitting element to all elements of the other transducer array, leading to a full 3D ultrasound tomography image of the breast.

Such a UST system 130 allows recording of volumetric ultrasound data, and the image resolution limited by slice thickness will be alleviated. In one exemplary design, the data acquisition electronics 18 allow a maximum of 768 parallel channels, so the number of transducers may be halved per array 132 a and 132 b. The coarser sampling in the plane of the array will be compensated by the cross illuminations

The scanner 130 of FIG. 11 can significantly improve image resolution and quality compared to those obtained from an ultrasound tomography system with one circular transducer array. A 3D ultrasound tomography system 10 of this configuration will be operator independent, which is critical for cancer screening, and will be more cost-effective than an ultrasound tomography system with a cylindrical transducer array.

F. Combination 2D Planar and 2D-Arc Array Scanner

FIG. 12 shows a scanner 140 comprising a semicircular or arcuate array 142 b having transducers 16 in an opposing or facing orientation with planar array 142 a, with target tissue 44 disposed between the two. The scanner 140 provides a combination of the advantages of the cylindrical transducer array 110 with those of the 2D planner array 100. An ultrasound tomography system 10 with such combination of transducer arrays improves the range of spatial coverage for data acquisition, and the planar array 142 can still be placed near the axillary region.

G. Combination 1D Beam and Arc Array Scanner

FIG. 13 illustrates a scanner 150 that reduces the 2D arrays in FIG. 12 to 1D arrays (arcuate line array 152 b and linear beam array 152 a). This configuration, using a one-dimeisional, straight-phased array 152 a and a 1D arc-shaped array, 152 reduces the number transducers 16, and thus the number of channels required for data acquisition electronics 18, while improving the spatial coverage of data acquisition compared to when using a two parallel phased transducer array scanner 12 in FIG. 5.

II. Synthetic Aperture Ultrasound Tomography Methods

Referring now to FIG. 14, a flow chart of a synthetic aperture ultrasound tomography method 200 is shown. This method is preferably used with any of the systems and scanners shown in FIG. 1 through FIG. 14, although other scanning systems are contemplated. Ideally, the method is used in conjunction with a scanner that has one or more arrays configured so that a plurality of transducers 16 of an array, or portion of an array, oppose (at a spaced-apart distance across the target scanning medium 44) a plurality of transducers 16 of either another portion of the array, or a separate array, so that reflection and transmission data may be acquired with each successive transducer excitation.

At step 202, the method performs a synthetic aperture ultrasound scan of the tissue medium in accordance with the schematic illustration of scanner 12 FIG. 3. At step 204, reflection and transission data are simultaneously acquired, as shown in the method 50 of FIG. 4. At step 206, ultrasound waveform tomagraphic imaging is performed on the acquired reflection and transmission data to generate a high-resolution ultrasound reconstruction image of the target medium 44.

As mentioned previously, a particular shortcoming of existing ultrasound omographic imaging is that they either use only transmission data, or reflection data only, for image reconstructions. In contrast, the synthetic-aperture ultrasound tomography method 200 of the present invention acquired both ultrasound transmission and reflection data at the same time, and use both ultrasound transmission and reflection data for tomographic reconstructions to greatly improve the shapes and quantitative values of mechanical properties of abnormalities.

FIGS. 15 through 18B demonstrate that using numerical breast-phantom data from ultrasound waveform tomography using both transmission and reflection data simultaneously significantly improves the accuracy of tomographic reconstructions, compared to those obtained using only ultrasound transmission data or only ultrasound reflection data.

Numerical phantom data was generated for a synthetic-aperture ultrasound tomography system with a two parallel phased transducer array scanner 12 as shown in FIG. 5. Each transducer array 14 a, 15 b is comprised of 384 evenly distributed ultrasound transducer elements, with a pitch size of 0.55 mm. The two transducer arrays were separated by 20 cm. The ultrasound source function used is a Ricker wavelet with a central frequency of 1.0 MHz.

FIG. 15 shows an image of a numerical breast phantom containing two different tumors (small, light tumor, and larger dark tumor). The background sound-speed of the phantom was 1500 ms, and those of the two tumor speeds were 1530 ms and 1550 ms, respectively. The diameters of the tumors were 2.0 mm and 7.0 mm, and approximately 1.3 wavelengths and 4.6 wavelengths. The two tumors were positioned along the longitudinal direction relative to the ultrasound transducer arrays. A high-order finite-difference time-domain wave-equation algorithm in accordance with step 206 was used to compute ultrasound transmission and reflection data.

FIG. 16A and FIG. 16B show imaging results (tomographic reconstruction in FIG. 16A, and vertical profile along the center of the tumors in FIG. 16B) obtained using only the reflection data. FIG. 17A and FIG. 17B show imaging results (tomographic reconstruction in FIG. 17A, and vertical profile along the center of the tumors in FIG. 17B) obtained using only the transmission data. FIG. 18A and FIG. 18B show imaging results (tomographic reconstruction in FIG. 18A, and vertical profile along the center of the tumors in FIG. 18B) obtained using both transmission and reflection data simultaneously in accordance with method 200.

The waveform tomographic reconstruction using only the reflection data (FIG. 16A and FIG. 16B) provides mostly the edge information of the tumors, and can distinguish the two tumors.

On the other hand, the waveform tomographic reconstruction (FIG. 17A and FIG. 17B) using only the transmission data gives mostly low spatial-wavenumber components of the tumors, and it is almost impossible to separate the two tumors.

By contrast, the waveform tomographic reconstruction using both the transmission and reflection data simultaneously (FIG. 18A and FIG. 18B) takes the advantages of the above two kinds of tomographic reconstructions, and produces an image with much improved tumor edges and sound-speed reconstructions.

A. Synthetic Aperture Ultrasound Waveform Tomography with Regularization

FIG. 19 illustrates a preferred method 206 for generating the ultrasound waveform step of method 200 (FIG. 14) using both transmission and reflection data for ultrasound waveform tomography. As shown in FIG. 19, reflection and transmission data are input at step 210, and ray approximation is performed at step 212 to generate an initial model. Next at step 214, image reconstruction is performed by computing the wave acoustic wave properties of the data by calculating the mean square difference between the observed and synthetic waveforms. In particular, step 214 is performed by performing iterative waveform inversion with regularization, as will be explained in further detail below. From a more basic level, performing step 214 is achieved by solving the acoustic wave equation of Eq. 1 with the minimization model of Eq. 5.

The acoustic-wave equation in the time-domain is given by:

$\begin{matrix} {{{\left\lbrack {{\frac{1}{K(r)}\frac{\partial^{2}}{\partial t^{2}}} - {\nabla{\cdot \left( {\frac{1}{\rho (r)}\nabla} \right)}}} \right\rbrack {p\left( {r,t} \right)}} = {s\left( {r,t} \right)}},} & {{Eq}.\mspace{14mu} 1} \end{matrix}$

where ρ(r) is the density, K(r) is the bulk modulus, s(t) is the source term, and p(r, t) is the pressure field.

The solution to Eq. 1 can be written as:

p=ƒ(K,s),  Eq. 2

where the function of ƒ is the forward modeling operator. Numerical techniques, such as finite-difference and spectral-element methods, can be used to solve Eq. 2. Let the model parameter be:

$m = {\begin{bmatrix} K \\ s \end{bmatrix}.}$

We can rewrite Eq. 2 as:

p=ƒ(m).  Eq. 3

For the case of constant density, Eq. 1 becomes:

$\begin{matrix} {{{\left( {{\frac{1}{C^{2}(r)}\frac{\partial^{2}}{\partial t^{2}}} - \nabla^{2}} \right){p\left( {r,t} \right)}} = {s\left( {r,t} \right)}},} & {{Eq}.\mspace{14mu} 4} \end{matrix}$

where C(r)=√{square root over (K(r)/ρ(r))}{square root over (K(r)/ρ(r))} is the sound speed, and the model parameter is m=C(r).

The inverse problem of Eq. 3 is posed as a minimization problem such that:

$\begin{matrix} {{{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{d - {f(m)}}}_{2}^{2} \right\}}},} & {{Eq}.\mspace{14mu} 5} \end{matrix}$

where ∥d−ƒ(m)∥₂ ² is the misfit function, d can be either ultrasound reflection data, or ultrasound transmission data, or combined ultrasound reflection and transmission data, and ∥·∥₂ stands for the L₂ norm. The minimization of Eq. 5 comprises solving for a model m that yields the minimum mean square difference between measured and synthetic waveforms.

However, because the measurements have limited coverage, solving Eq. 5 is ill-posed. Moreover, because of the nonlinearity of the function ƒ, the solution can be trapped in the neighborhood of a local minimum of the misfit function. Therefore, a regularization technique is preferably applied to Eq. 5 to alleviate the ill-posedness of the inversion.

A general form of regularization is often written as:

$\begin{matrix} {{{E(m)} = {\min\limits_{m}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda \; {R(m)}}} \right\}}},} & {{Eq}.\mspace{14mu} 6} \end{matrix}$

where R(m) is the regularization term.

The methods of the present invention are directed to performing ultrasound waveform tomography of acquired reflection and transmission signals with use of a regularization scheme. In particular, transmission and reflection data are used for ultrasound waveform tomography with a total variation (TV) regularization scheme and modified TV regularization scheme.

Similar to the experiments conducted with respect to FIG. 15 through FIG. 18B, testing was also done to determine the advantage of using both transmission and reflection data for ultrasound waveform tomography with a regularization scheme. Ultrasound waveform tomography with a modified total-variation regularization scheme (discussed in further detail below) to the same numerical phantom data. FIG. 20A through FIG. 20B, FIG. 21A through FIG. 21B, and FIG. 22A through FIG. 22B, respectively, show tomographic reconstruction results obtained using only reflection data, only transmission data, and both transmission and reflection data. Ultrasound waveform tomography with the modified total-variation regularization significantly reduces the image artifacts, and improves the accuracy of sound-speed reconstructions. However, the tomographic reconstruction in FIG. 20A through FIG. 20B, obtained using only the reflection data, still gives mostly the edge information for the larger tumor. The tomographic reconstruction in FIG. 21A through FIG. 21B, produced using only the transmission data, cannot separate the two tumors. When using both the transmission and reflection data simultaneously, our ultrasound waveform tomography with the modified total-variation regularization gives accurate reconstructions of the shapes and the sound-speeds of both tumors.

For each of the TV regularization and modified TV regularization ultrasound waveform tomography methods of the present invention, the following description will give an overview on the derivation of the equations used to solve the particular image reconstruction, followed by a discussion of algorithms and computations methods of the present invention for generating the reconstructed images. Then, numerical results for tests showing the efficacy of both methods will be described and illustrated.

B. Synthetic Aperture Ultrasound Waveform Tomography with TV Regularization.

The Tikhonov regularization can be formulated as

$\begin{matrix} {{{E(m)} = {\min\limits_{m}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda {{Hm}}_{2}^{2}}} \right\}}},} & {{Eq}.\mspace{14mu} 7} \end{matrix}$

where the matrix H is usually defined as a high-pass filtering operator, or an identity matrix. The Tikhonov regularization is a L₂-norm-based regularization; therefore, it is best suited for the smooth parameter m. In other words, the Tikhonov regularization is not the best option for sound-speed reconstructions of small breast tumors.

The TV regularization can be written as:

$\begin{matrix} {{{E(m)} = {\min\limits_{m}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda {m}_{TV}}} \right\}}},} & {{Eq}.\mspace{14mu} 8} \end{matrix}$

where the TV regularization for 2D model is defined as the L₁ norm:

$\begin{matrix} {{{{m}_{TV} = {{{\nabla m}}_{1} = {\sum\limits_{{1 \leq i},{j \leq n}}\; \sqrt{{\left( {\nabla_{x}m} \right)_{i,j}}^{2} + {\left( {\nabla_{z}m} \right)_{i,j}}^{2}}}}},{{{with}\left( {\nabla_{x}m} \right)}_{i,j} = {m_{{i + 1},j} - m_{i,j}}}}{{{and}\left( {\nabla_{z}m} \right)}_{i,j} = {m_{i,{j + 1}} - {m_{i,j}.}}}} & {{Eq}.\mspace{14mu} 9} \end{matrix}$

For sound-speed reconstructions of small breast tumors, the methods of the present invention exploit the TV regularization for performing ultrasound waveform tomography using both transmission and reflection data. Because of the nonlinearity of the TV term in Eq. 9 and its non-differentiability at the origin, the method of the present invention is also directed to approximate the original TV functional so that it can be differentiated. A preferred approach is to approximate the TV term as:

$\begin{matrix} {{{m}_{{TV},\varepsilon} = {\sum\limits_{{1 \leq i},{j \leq n}}\; \sqrt{{\left( {\nabla_{x}m} \right)_{i,j}}^{2} + {\left( {\nabla_{z}m} \right)_{i,j}}^{2} + \varepsilon}}},} & {{Eq}.\mspace{14mu} 10} \end{matrix}$

where ε is a small positive constant, making ∥m∥_(TV,ε) differentiable, so the gradient and Hessian can be calculated.

The role of the positive constant ε is to approximate the Euclidean norm |x| with a smooth function √{square root over (|x²|+ε)}. The larger the value of ε, the smoother the approximation. Whereas the smaller the value of ε, the closer the approximation to the original function, and the more singular the approximation becomes at the origin. Therefore, we need to use a reasonable value for ε, which strikes a balance between a good approximation and the stability. The value of ε was set to be 0.1 in the numerical tests detailed below.

The other parameter used in the TV regularization function (Eq. 10) of the present invention is λ, which is configured to balance between the data fidelity term ∥d−f(m)∥₂ ² and the regularization term ∥m∥_(TV). Previous methods to automatically pick the parameter have been for linear applications. Because of the nonlinearity of the problem, those methods cannot be directly applied to the TV regularization function (Eq. 10) of the present invention.

Accordingly, one method of the present invention is to pick λ, as a function of Eq. 11:

$\begin{matrix} {{\lambda = \frac{{{d - {f(m)}}}_{2}^{2}}{r \cdot {m}_{TV}}},} & {{Eq}.\mspace{14mu} 11} \end{matrix}$

where the value of r is defined to be the ratio between the data fidelity term and the regularization term. A value of 1×10⁷<r<1×10⁸ r was found to return reasonable results.

There are two major computational methods involved in solving for the minimization problem of Eq. 5: the forward modeling algorithm and the inverse optimization algorithm. Because the model parameters for the model m that we are solving have high dimensionality, an iterative optimization method based on the Krylov subspace is used, or Nonlinear Conjugate Gradient (NCG).

One critical step in the optimization is the calculation of the measurements for given model parameters, which is usually called forward modeling. It is usually the most computationally intensive. Here, for the forward modeling, the finite-difference method with staggered grids in both space and time domain was used.

Instead of using the wave equation, which is a second-order hyperbolic equation, we use the equivalent first-order equations for illustration:

$\quad\begin{matrix} \left\{ \begin{matrix} {\frac{\partial p}{\partial t} + {{K(r)}\left( {\frac{\partial u}{\partial x} + \frac{\partial v}{\partial z}} \right)}} & {{= {s\left( {r,t} \right)}},} \\ {\frac{\partial u}{\partial t} + {\frac{1}{\rho (r)}\frac{\partial p}{\partial x}}} & {{= 0},} \\ {\frac{\partial v}{\partial t} + {\frac{1}{\rho (r)}\frac{\partial p}{\partial z}}} & {{= 0},} \end{matrix} \right. & {{Eq}.\mspace{14mu} 12} \end{matrix}$

where u is the wavefield velocity along the x direction, and v is the wavefield velocity along the z direction. If a two-point central finite-difference scheme is used to approximate the derivative in the time domain, and a fourth-order Taylor expansion is used to interpolate for the derivative values in the space domain, this finite-difference scheme yields a fourth-order accuracy in space and a second-order accuracy in time. The superscripts and subscripts are meant to distinguish between increments in time and space:

$\begin{matrix} {\quad\left\{ {\begin{matrix} {{p_{i,{j + {1\text{/}2}}}^{k + {1\text{/}2}} = {p_{i,{j + {1\text{/}2}}}^{k - {1\text{/}2}} + {K_{i,{j + {1\text{/}2}}}{\frac{\delta \; t}{\delta \; x}\left\lbrack {\left( \frac{\partial u}{\partial x} \right)_{i,{j + {1\text{/}2}}}^{k - {1\text{/}2}} + \left( \frac{\partial v}{\partial z} \right)_{i,{j + {1\text{/}2}}}^{k - {1\text{/}2}}} \right\rbrack}} + {s\left( {r,t} \right)}}},} \\ {{u_{{i + {1\text{/}2}},{j + {1\text{/}2}}}^{k + {1\text{/}2}} = {u_{{i + {1\text{/}2}},{j + {1\text{/}2}}}^{k - {1\text{/}2}} + {\frac{1}{\rho_{{i + {1\text{/}2}},{j + {1\text{/}2}}}}\left( \frac{\partial p}{\partial x} \right)_{{i + {1\text{/}2}},{j + {1\text{/}2}}}^{k - {1\text{/}2}}}}},} \\ {{v_{i,j}^{k + {1\text{/}2}} = {v_{i,j}^{k - {1\text{/}2}} + {\frac{1}{\rho_{i,j}}\left( \frac{\partial p}{\partial z} \right)_{i,j}^{k - {1\text{/}2}}}}},} \end{matrix}{where}} \right.} & {{Eq}.\mspace{14mu} 13} \\ {\quad\left\{ \begin{matrix} {{\left( \frac{\partial u}{\partial x} \right)_{i,{j + {1\text{/}2}}} = {{0.0417\left( {u_{{i - {3\text{/}2}},{j + {1\text{/}2}}} - u_{{i + {3\text{/}2}},{j + {1\text{/}2}}}} \right)} + {1.125\left( {u_{{i - {1\text{/}2}},{j + {1\text{/}2}}} - u_{{i + {1\text{/}2}},{j + {1\text{/}2}}}} \right)}}},} \\ {{\left( \frac{\partial v}{\partial z} \right)_{i,{j + {1\text{/}2}}} = {{0.0417\left( {v_{i,{j - 1}} - v_{i,{j + 2}}} \right)} + {1.125\left( {v_{i,j} - v_{i,{j + 1}}} \right)}}},} \\ {{\left( \frac{\partial p}{\partial x} \right)_{{i + {1\text{/}2}},{j + {1\text{/}2}}} = {{0.0417\left( {p_{{i - 1},{j + {1\text{/}2}}} - p_{{i + 2},{j + {1\text{/}2}}}} \right)} + {1.125\left( {p_{i,{j + {1\text{/}2}}} - p_{{i + 1},{j + {1\text{/}2}}}} \right)}}},} \\ {\left( \frac{\partial p}{\partial z} \right)_{i,j} = {{0.0417\left( {p_{i,{j - {3\text{/}2}}} - p_{i,{j + {3\text{/}2}}}} \right)} + {1.125{\left( {p_{i,{j - {1\text{/}2}}} - p_{i,{j + {1\text{/}2}}}} \right).}}}} \end{matrix} \right.} & {{Eq}.\mspace{14mu} 14} \end{matrix}$

NCG was used to solve the optimization problem in Eq. 8. The gradient of the cost function is needed for computing the search direction. The gradient of the cost function E(m) is needed for computing the search direction. The gradient of the data misfit can be obtained using the adjoint-state method:

$\begin{matrix} {{{\nabla_{m}{{d - {f(m)}}}_{2}^{2}} = {2{\rho \left( {C^{(k)}(r)} \right)}^{2}{\sum\limits_{shots}\; {\sum\limits_{t}\; {{\nabla{\cdot {{\overset{->}{u}}^{(k)}\left( {r,t} \right)}}}{\nabla{\cdot {{\overset{\leftarrow}{b}}^{(k)}\left( {r,t} \right)}}}}}}}},} & {{Eq}.\mspace{14mu} 15} \end{matrix}$

where {right arrow over (u)}^((k)) is the forward propagated wavefield, and

^((k)) is the backward propagated residual at iteration k, which is defined as r^((k))=d^(obs)−f(m^((k)))

The gradient of the TV term is obtained using:

$\begin{matrix} {{{\nabla_{m}{{TV}(m)}} = {{{\left\lbrack {D_{x}^{T}\mspace{14mu} D_{z}^{T}} \right\rbrack \begin{bmatrix} {{diag}\left( {\varphi^{\prime}(m)} \right)} & 0 \\ 0 & {{diag}\left( {\varphi^{\prime}(m)} \right)} \end{bmatrix}}\begin{bmatrix} D_{x} \\ D_{z} \end{bmatrix}}m}},} & {{Eq}.\mspace{14mu} 16} \end{matrix}$

where φ(t)=√{square root over (t²+ε)}, φ′(m)=φ′(D_(x)m²+D_(z)m²) and the spatial derivative D_(x)m=m_(i+1,j)−m_(i,j) and D_(z)m=m_(i,j+1)−m_(i,j) for the spatial grid indices i and j.

Therefore, the gradient of the cost function E(m) is:

∇_(m) E(m)=∇_(m) ∥d−f(m)∥₂ ²+∇_(m) TV(m).  Eq. 17

The search direction q_((k)) at iteration k is then defined to be the conjugate to the gradient at the current iteration step. Once the search direction q^((k)) at iteration k is obtained, the line search with the following Armijo criteria is further used for the optimal step size β^((k)):

$\begin{matrix} {\quad\left\{ \begin{matrix} {E\left( {m^{(k)} + {\beta^{(k)}q^{(k)}}} \right)} & {{\leq {{E\left( m^{(k)} \right)} + {c_{1}{\beta^{(k)}\left( q^{(k)} \right)}^{T}{\nabla{E\left( m^{(k)} \right)}}}}},} \\ {\left( q^{(k)} \right)^{T}{\nabla{E\left( {m^{(k)} + {\beta^{(k)}q^{(k)}}} \right)}}} & {\geq {{c_{2}\left( q^{(k)} \right)}^{T}{{\nabla{E\left( m^{(k)} \right)}}.}}} \end{matrix} \right.} & {{Eq}.\mspace{14mu} 18} \end{matrix}$

When the search direction q^((k)) and the step size β^((k)) are determined, the update of the current iteration is updated according to Eq. 19:

m ^((k+)1)=m ^((k))+β^((k)) q ^((k))  Eq. 19

FIG. 23 illustrates a flow diagram for a method 220 for performing the computing step 214 of FIG. 19. This is achieved via solving the acoustic wave (Eq. 1) by using the NCG based algorithm (Algorithm 1 below) to approximate the TV regularization term (Eq. 8) according to Eq. 10.

Referring to FIG. 23, the first step in the method 220 is to input the specified tolerance TOL, in addition to the initial model m⁽⁰⁾. Other parameters, such as λ, which is preferably determined according to the method of the present invention shown in Eq. 11, may also be input in this step. The initial model m⁽⁰⁾ may be generated via applying the ray approximation step 212 of the input reflection and transmission data 210, as shown in FIG. 19.

At step 224, the parameters are initialized (e.g. the current iteration value k is set at zero).

At step 226, the algorithm queries whether the current iteration of the model has met the minimum value set by the assigned tolerance TOL.

If the threshold value has not been met, the algorithm computes the step size by computing Eq. 18 at step 228.

Next, at step 230, the current iteration model m^((k)) is updated based on step size β^((k)) and search direction q^((k)) according to Eq. 19.

At step 332, the gradient of the cost function ∇E^((k+1)) is computed according to Eq. 17.

At step 234, the ratio of the inner product of the gradient VE is computed to find the term γ^((k+1)) according to:

$\gamma^{({k + 1})} = {\frac{{< {\nabla E^{({k + 1})}}},{{\nabla E^{({k + 1})}} >}}{{< {\nabla E^{(k)}}},{{\nabla E^{(k)}} >}}.}$

Finally, the search direction q^((k)) is updated at step 236 according to:

q ^((k+1)) =−∇E ^((k+1))+γ^((k+1)) q ^((k))

The current iteration value k is then updated and the process repeated at step 238.

If the threshold tolerance has been met at step 226, then the process ends and outputs the model m^((k)) at step 240. If not, the process continues to iterate until it does.

Algorithm 1: Canonical NCG to solve min_(m) E(m) in Eq. 8 Input: m⁽⁰⁾, TOL Output: m^((k)) 1: Initialize k = 0, E⁽⁰⁾ = E(m⁽⁰⁾), ∇E⁽⁰⁾ = ∇E(m⁽⁰⁾), and q⁽⁰⁾ = ∇E⁽⁰⁾; 2: while {||∇E^((k))|| > TOL} do 3:  Compute the step size β^((k)) satisfying Eq. 18; 4:  Update the solution according to Eq. 19; 5:  Compute the gradient ∇E^((k+1)) based on Eq. 17; 6:  Compute the ratio of the inner product of the gradient ∇E :   ${\gamma^{({k + 1})} = \frac{\langle{{\nabla E^{({k + 1})}},{\nabla E^{({k + 1})}}}\rangle}{\langle{{\nabla E^{(k)}},{\nabla E^{(k)}}}\rangle}};$ 7:  Update the search direction: q^((k+1)) = −∇E^((k+1)) + γ^((k+1))q^((k)); 8:  Next iteration: k ← k + 1; 9: end while

C. Synthetic Aperture Ultrasound Waveform Tomography with Modified TV Regularization.

The conventional TV regularization discussed above suffers from slow convergence. Thus, a modified TV regularization scheme may to be adapted into ultrasound waveform tomography.

The cost function with the modified TV regularization is given by:

$\begin{matrix} {{{E\left( {m,u} \right)} = {\min\limits_{m,u}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda_{1}{{m - u}}_{2}^{2}} + {\lambda_{2}{{\nabla u}}_{1}}} \right\}}},} & {{Eq}.\mspace{14mu} 20} \end{matrix}$

where λ₁ and λ₂ are both positive regularization parameters, and an auxiliary variable u is added to the conventional TV functional in Eq. 8.

Hence, the modified TV regularization in Eq. 20 can be equivalently written as

$\begin{matrix} {{{E\left( {m,u} \right)} = {\min\limits_{u}\left\{ {{\min\limits_{m}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda_{1}{{m - u}}_{2}^{2}}} \right\}} + {\lambda_{2}{{\nabla u}}_{1}}} \right\}}},} & {{Eq}.\mspace{14mu} 21} \end{matrix}$

An alternating-minimization algorithm can therefore be employed for solving the double minimization problem in Eq. 21. Beginning with a starting model u⁽⁰⁾, solving for Eq. 21 leads to the solutions of two minimization problems:

$\begin{matrix} {\quad\underset{\square}{\begin{matrix} \left\{ \begin{matrix} {m^{(k)} = {{\underset{m}{\arg \; \min}{{d - {f(m)}}}_{2}^{2}} + {\lambda_{1}{{m - u^{({k - 1})}}}_{2}^{2}}}} \\ {u^{(k)} = {{\underset{u}{\arg \; \min}{{m^{(k)} - u}}_{2}^{2}} + {\lambda_{2}{{\nabla u}}_{1}}}} \end{matrix} \right. & {{Eq}.\mspace{14mu} 22} \end{matrix}}} & {{Eq}.\mspace{14mu} 22} \end{matrix}$

for i=1, 2, . . . . We describe the computational methods for solving these two minimization problems in Eq. 22. Compared to the conventional TV function in Eq. 8, the benefits of solving the modified TV functional by adding the auxiliary variable u in Eq. 22 are twofold. First, we simplify the complexity of the original optimization problem of nonlinear inversion with the TV constraint. From Eq. 22, these two sub-optimization problems have distinct physical meanings: solving for m^((k)) is the nonlinear inversion with a Tikhonov regularization constraint; solving for u^((k)) is to preserve the edges. Therefore, the interleaving of solving these two variables leads to a solution minimizing the data misfit term with edges preserved. The other benefit that we gain out of solving Eq. 22 is that solving for u^((k)) is a typical L₁-TV problem, and we can smartly choose certain computational methods, such that the selection of the smoothing parameter of ε in Eq. 10 can be avoided.

Selecting the right values for the parameters used in the conventional and modified TV regularization schemes is important for the effective use of these regularization schemes.

The role of the positive constant E is to approximate the Euclidean norm ∥x∥ with a smooth function √{square root over (∥x²∥+ε)} as in Eq. 10. The larger the value of ε, the smoother the approximation, whereas the smaller the value of ε, the closer the approximation to the original function, and the more singular the approximation becomes at the origin. Therefore, we need to use a reasonable value for ε, which strikes a balance between a good approximation and stability. Otherwise, an inappropriate selection of ε not only can result in inaccurate results, but also can significantly increase the number of iterations required. The following method of the present invention the is used to solve the optimization problem and allows us to avoid the selection of this smoothing parameter ε.

The other parameters used in Eq. 22 are λ₁ and λ₂. From the above formulation, the meaning of two regularization parameters can be specifically determined. The parameter λ₁ controls the trade-off between the data misfit term ∥d−ƒ(m)∥₂ ² and the regularization term ∥m−u^((k−1))∥₂ ², and λ₂ controls the amount of edge-preservation to the reconstruction. Existing methods all rely on an explicitly constructed forward operator A, such that the trace value of A(A^(T)+λI)⁻¹ A^(T) is required to proceed the estimation of the regularization parameters.

In the method of the present invention, λ₁ and λ₂ are selected such that:

$\begin{matrix} {{\lambda_{i} = \frac{{data}\mspace{14mu} {fidelity}\mspace{14mu} {term}}{{r_{i} \cdot {regularization}}\mspace{14mu} {term}}},{i = {1,2,}}} & {{Eq}.\mspace{14mu} 23} \end{matrix}$

where the values of r_(i) are defined to be the ratio between the data fidelity term and the regularization term. We find that 1×10⁷<r₁<1×10⁸, and 0.1<r₂<1 usually yield reasonably good reconstruction results.

Solving for the minimization problem in Eq. 20 is decomposed into solving two separate minimization problems:

$\begin{matrix} {{{E(m)} = {{\min\limits_{m}{{d - {f(m)}}}_{2}^{2}} + {\lambda_{1}{{m - u^{({k - 1})}}}_{2}^{2}}}},{and}} & {{Eq}.\mspace{14mu} 24} \\ {{E(u)} = {{\min\limits_{u}{{u - m^{(k)}}}_{2}^{2}} + {\lambda_{2}{{u}_{TV}.}}}} & {{Eq}.\mspace{14mu} 25} \end{matrix}$

Therefore, a sequence of iterations are generated, specifically

u ⁽⁰⁾ ,m ⁽¹⁾ ,u ⁽¹⁾ ,m ⁽²⁾ ,u ⁽²⁾ ,m ⁽³⁾ ,u ⁽³⁾ , . . . ,m ^((k)) ,u ^((k)), . . . .

which is shown in more detail in the computational method 250 of FIG. 24 for solving Eq. 20, and is embodied in Algorithm 2 below.

In the following, the computational methods for solving Eq. 20 and the two minimization problems of Eq. 24 and Eq. 25 in accordance with the present invention are shown with reference to FIG. 24, FIG. 25, and FIG. 26, in addition to Algorithms 2, 3, and 4 below.

Referring to FIG. 24, the computational method 250 for solving Eq. 20 comprises first inputting the specified tolerance TOL, in addition to the initial model u⁽⁰⁾ at step 252. Other parameters, such as λ, which is preferably determined according to the method shown in Eq. 23 of the present invention, may also be input in this step. The initial model u⁽⁰⁾ may be generated via applying ray approximation step 212 on the input reflection and transmission data 210 as shown in FIG. 19.

At step 254, the parameters are initialized (e.g. the current iteration value k is set at zero).

A step 256, the algorithm queries whether the current iteration of the model has met the minimum value set by the assigned tolerance TOL.

If the threshold value has not been met, the algorithm solves Eq. 24 to compute m^((k)) according to Algorithm 3 and method 270 of FIG. 5 at step 258.

At step 260, u^((k)) is computed according to Algorithm 4 and method 300 of FIG. 26. The current iteration value is then updated at step 262, and the routine returns to step 256. If the threshold is met, the algorithm ends and outputs u^((k)) at step 264. If not, Algorithm 2 continues to iterate.

Algorithm 2 Computational method for solving Eq. 20 Input: u⁽⁰⁾ , TOL Output: u^((k)) 1: Initialize k = 0 ; 2: while { ∥ m^((k)) − m^((k−1)) ∥> TOL } do 3: Solve Eq. 24 for m^((k)) according to Alg. 3; 4: Solve Eq. 25 for u^((k)) according to Alg. 4; 5: Next iteration: k ← k + 1 ; 6: end while

The minimization problem of Eq. 24 (step 258 of FIG. 24) is a conventional Tikhonov-type nonlinear-regularization problem. Accordingly, a NCG is used as the optimization method. To use NCG for solving Eq. 24, the gradient of the cost function is required to compute the search direction. The gradient of the data misfit can be obtained by the adjoint-state method:

$\begin{matrix} {{{\nabla_{m}{{d - {f(m)}}}_{2}^{2}} = {2{\rho \left( {C^{(k)}(r)} \right)}^{2}{\sum\limits_{shots}\; {\sum\limits_{t}\; {{\nabla{\cdot {{\overset{->}{u}}^{(k)}\left( {r,t} \right)}}}{\nabla{\cdot {{\overset{\leftarrow}{b}}^{(k)}\left( {r,t} \right)}}}}}}}},} & {{Eq}.\mspace{14mu} 26} \end{matrix}$

where {right arrow over (u)}^((k)) is the forward propagated wavefield, and

^((k)) is the backward propagated residual at iteration k, which is further defined as r^((k))=d^(obs)−ƒ(m^((k))).

Therefore, the gradient of the cost function E(m) is:

$\begin{matrix} {{\nabla_{m}{E(m)}} = {{2{\rho \left( {C^{(k)}(r)} \right)}^{2}{\sum\limits_{shots}\; {\sum\limits_{t}\; {{\nabla{\cdot {{\overset{->}{u}}^{(k)}\left( {r,t} \right)}}}{\nabla{\cdot {{\overset{\leftarrow}{b}}^{(k)}\left( {r,t} \right)}}}}}}} + {2{{\lambda_{1}\left( {m - u^{({i - 1})}} \right)}.}}}} & {{Eq}.\mspace{14mu} 27} \end{matrix}$

The search direction q^((k)) at iteration k is then defined to be the conjugate to the gradient at the current iteration step. Once the search direction q^((k)) at iteration k is obtained, the line search with the Armijo criteria below is further used for the optimal step size β^((k)):

$\quad\begin{matrix} \left\{ \begin{matrix} {E\left( {m^{(k)} + {\beta^{(k)}q^{(k)}}} \right)} & {\leq {{E\left( m^{(k)} \right)} + {c_{1}{\beta^{(k)}\left( q^{(k)} \right)}^{T}{\nabla{E\left( m^{(k)} \right)}}}}} \\ {\left( q^{(k)} \right)^{T}{\nabla{E\left( {m^{(k)} + {\beta^{(k)}q^{(k)}}} \right)}}} & {\geq {{c_{2}\left( q^{(k)} \right)}^{T}{{\nabla{E\left( m^{(k)} \right)}}.}}} \end{matrix} \right. & {{Eq}.\mspace{14mu} 28} \end{matrix}$

With the search direction q^((k)) and the step size β^((k)) determined, the update of the current iteration is therefore given by:

m ^((k+1)) =m ^((k))+β^((k)) q ^((k))  Eq. 29

The NCG algorithm 270 shown in FIG. 25, used for solving Eq. 24, is also in Algorithm 3. The first step in the method 270 is to input the specified tolerance TOL, in addition to the initial model m⁽⁰⁾. Other parameters, such as λ₁ and λ₂, which are preferably determined according to the method of the present invention shown in Eq. 23, may also be input in this step. The initial model m⁽⁰⁾ may be generated via applying ray approximation step 212 of the input reflection and transmission data 210 as shown in FIG. 19.

At step 274, the parameters are initialized (e.g. the current iteration value k is set at zero).

A step 276, the algorithm queries whether the current iteration of the model has met the minimum value set by the assigned tolerance TOL.

If the threshold value has not been met, the algorithm computes the step size by computing Eq. 28 at step 278.

Next, at step 280, the current iteration model m^((k)) is updated based on step size β^((k)) and search direction q^((k)) according to Eq. 29.

At step 282, the gradient of the cost function ∇E^((k+1)) is computed according to Eq. 27.

At step 284, the ratio of the inner product of the gradient VE is computed to find the term γ^((k+1)) according to:

$\gamma^{({k + 1})} = {\frac{{< {\nabla E^{({k + 1})}}},{{\nabla E^{({k + 1})}} >}}{{< {\nabla E^{(k)}}},{{\nabla E^{(k)}} >}}.}$

Finally, the search direction q^((k)) is updated at step 286 according to:

q ^((k+1)) =−∇E ^((k+1))+γ^((k+1)) q ^((k))

The current iteration value k is then updated at step 288, and the process repeated at step 276.

If the threshold tolerance has been met at step 276, then the process ends, and outputs the model m^((k)) at step 290. If not, the process continues to iterate until it does.

${Algorithm}\mspace{14mu} 3\text{:}\mspace{14mu} {Canonical}\mspace{14mu} {NCG}\mspace{14mu} {to}\mspace{14mu} {solve}\mspace{14mu} {\min\limits_{m}\; {E(m)}}$ Input: m⁽⁰⁾, TOL Output: m^((k)) 1: Initialize k = 0, E⁽⁰⁾ = E(m⁽⁰⁾), ∇E⁽⁰⁾ = ∇E(m⁽⁰⁾), and q⁽⁰⁾ = ∇E⁽⁰⁾; 2: while {||∇E^((k))|| > TOL} do 3:  Compute the step size β^((k)) satisfying Eq. (28); 4:  Update the solution according to Eq. (29); 5:  Compute the gradient ∇E^((k+1)) based on Eq. (27); 6:  Compute the ratio of the inner product of the gradient ∇E :   ${\gamma^{({k + 1})} = \frac{\langle{{\nabla E^{({k + 1})}},{\nabla E^{({k + 1})}}}\rangle}{\langle{{\nabla E^{(k)}},{\nabla E^{(k)}}}\rangle}};$ 7:  Update the search direction: q^((k+1)) = −∇E^((k+1)) + γ^((k+1))q^((k)); 8:  Next iteration: k ← k + 1; 9: end while

While there are many numerical methods for solving the L₂-TV problem describing in Eq. 25, the split-Bregman method approach was found to be appropriate. Optimization problems of the L₁-TV or L₂-TV type are most efficiently solved by using the Bregman distance, which is defined as

D _(E) ^(p)(u,v)=E(u)−E(v)−<p,u−v>,  Eq. 30

where p is the subgradient of E at v. The basic idea of the split-Bregman method is to reformulate Eq. 25 as an equivalent minimization problem based on the Bregman distance:

$\begin{matrix} {{{{E\left( {u,g_{x},g_{z}} \right)} = {{\min\limits_{u,g_{x},g_{z}}{{u - m^{(k)}}}_{2}^{2}} + {\lambda_{2}{u}_{TV}} + {\mu {{g_{x} - {\nabla_{x}u} - b_{x}^{(k)}}}_{2}^{2}} + {\mu {{g_{z} - {\nabla_{z}u} - b_{z}^{(k)}}}_{2}^{2}}}},\mspace{79mu} {where}}{{g_{x}^{(k)} = {\nabla_{x}u^{(k)}}},{g_{z}^{(k)} = {\nabla_{z}u^{(k)}}},{b_{x}^{({k + 1})} = {b_{x}^{(k)} + \left( {{\nabla_{x}u^{({k + 1})}} - g_{x}^{({k + 1})}} \right)}},\mspace{79mu} {and}}\mspace{79mu} {b_{z}^{({k + 1})} = {b_{z}^{(k)} + {\left( {{\nabla_{z}u^{({k + 1})}} - g_{z}^{({k + 1})}} \right).}}}} & {{Eq}.\mspace{14mu} 31} \end{matrix}$

To solve this minimization problem, an alternating minimization algorithm is employed to Eq. 31, where two subproblems need to be further minimized:

$\begin{matrix} {{{E(u)} = {{\min\limits_{u}{{u - m^{(k)}}}_{2}^{2}} + {\mu {{g_{x}^{(k)} - {\nabla_{x}u} - b_{x}^{(k)}}}_{2}^{2}} + {\mu {{g_{z}^{(k)} - {\nabla_{z}u} - b_{z}^{(k)}}}_{2}^{2}}}},\mspace{79mu} {and}} & {{Eq}.\mspace{14mu} 32} \\ {{E\left( {g_{x},g_{z}} \right)} = {{\min\limits_{g_{x},g_{z}}\mspace{14mu} {\lambda_{2}{u}_{TV}}} + {\mu {{g_{x} - {\nabla_{x}u} - b_{x}^{(k)}}}_{2}^{2}} + {\mu {{{g_{z} - {\nabla_{z}u} - b_{z}^{(k)}}}_{2}^{2}.}}}} & {{Eq}.\mspace{14mu} 33} \end{matrix}$

Eq. 32 satisfies the optimality condition:

(I−μΔ)u ^((k+1)) =m ^((k))+μ∇_(x) ^(T)(g _(x) ^((k)) −b _(x) ^((k)))+μ∇_(z) ^(T)(g _(z) ^((k)) −b _(z) ^((k)))  Eq. 34

The solution of this equation is obtained using the Gauss-Seidel iterative method:

$\begin{matrix} {u_{i,j}^{(k)} = {{\frac{\mu}{1 + {4\mu}}\left( {u_{{i + 1},j}^{(k)} + u_{{i - 1},j}^{(k)} + u_{i,{j + 1}}^{(k)} + u_{i,{j - 1}}^{(k)} + g_{x,{i - 1},j}^{(k)} - g_{x,i,j}^{(k)} + g_{z,i,{j - 1}}^{(k)} - g_{z,i,j}^{(k)} - b_{x,{i - 1},j}^{(k)} + b_{x,i,j}^{(k)} - b_{z,i,{j - 1}}^{(k)} + b_{y,i,j}^{(k)}} \right)} + {\frac{1}{1 + {4\mu}}{m_{i,j}^{(k)}.}}}} & {{Eq}.\mspace{14mu} 35} \end{matrix}$

Equation 33 is solved explicitly using a generalized shrinkage formula:

$\begin{matrix} {{g_{x}^{({k + 1})} = {{\max \left( {{s^{(k)} - {\lambda_{2}\text{/}2\mu}},0} \right)}\; \frac{{\nabla_{x}u^{(k)}} + b_{x}^{(k)}}{s^{(k)}}}},{and}} & {{Eq}.\mspace{14mu} 36} \\ {{{g_{z}^{({k + 1})} = {\max \; \left( {{s^{(k)} - {\lambda_{2}\text{/}2\mu}},0} \right)\; \frac{{\nabla_{z}u^{(k)}} + b_{z}^{(k)}}{s^{(k)}}}},{where}}{s^{(k)} = {\sqrt{{{{\nabla_{x}u^{(k)}} + b_{x}^{(k)}}}^{2} + {{{\nabla_{z}u^{(k)}} + b_{z}^{(k)}}}^{2}}.}}} & {{Eq}.\mspace{14mu} 37} \end{matrix}$

The numerical algorithm 300 of FIG. 26 for solving Eq. 31 using the split-Bregman iteration scheme is also provided in Algorithm 4.

The first step in the method 300 is to input the specified tolerance TOL, in addition to the initial model in addition to the initial model u⁽⁰⁾ at step 302. The initial model u⁽⁰⁾ may be generated via applying ray approximation step 212 on the input reflection and transmission data 210 as shown in FIG. 19.

At step 304, the parameters are initialized (e.g. the current iteration value k is set at zero).

At step 306, the algorithm queries whether the current iteration of the model has met the minimum value set by the assigned tolerance TOL.

If the threshold value has not been met, the algorithm solves Eq. 32 using the Gauss-Seidel equation (Eq. 35) at step 308.

At step 310, Eq. 32 is then solved using generalized shrinkage (Eq. 36 and Eq. 37).

Next, the intermediate variable b_(x) is updated according to b_(x) ^((k+1))=b_(x) ^((k))+(∇_(x)u^((k+1))−g_(x) ^((k+1))) at step 312.

The intermediate variable b_(z) is then updated according to b_(x) ^((k+1))=b_(x) ^((k))+(∇_(z)u^((k+1))−g_(x) ^((k+1))) at step 314.

The current iteration value is then updated at step 316, and the routine returns to step 306. If the threshold is met, the algorithm ends and outputs u^((k)) at step 318. If not, Algorithm 4 continues to iterate.

Algorithm 4: Split-Bregman iteration Input: u⁽⁰⁾ , TOL Output: u^((k)) 1: Initialize k = 0 , g_(x) ⁽⁰⁾ = g_(z) ⁽⁰⁾ = b_(x) ⁽⁰⁾ = b_(z) ⁽⁰⁾ = 0 ; 2: while { ∥ u^((k)) − u^((k−1)) ∥> TOL } do 3: Solve Eq. (32) according to Eq. (35); 4: Solve Eq. (33) according to Eq. (36) and Eq. (37); 5: Update b_(x) ^((k+1)) = b_(x) ^((k)) + (∇_(x)u^((k+1)) − g_(x) ^((k+1))) ; 6: Update b_(z) ^((k+1)) = b_(z) ^((k)) + (∇_(z)u^((k+1)) − g_(z) ^((k+1))) ; 7: Next iteration: k ← k + 1 ; 8: end while

To better understand the computational methods in Algorithm 2, it is worthwhile to calculate the computational costs of each of its steps.

Let the size of the model be mε

^({tilde over (m)}×ñ) and the data pε

^({tilde over (q)}×ñ) where {tilde over (m)} is the depth, ñ is the offset, and {tilde over (q)} is the time steps. Assuming that there are s sources and the finite-difference calculation employs a scheme of o(δt², δh⁴). The total cost of Alg. 2 for one iteration step is the sum of the cost from Alg. 3 and the cost from Alg. 4. By adding the costs from all the steps in Alg. 3, the approximated cost for each iteration step is:

COST₁≈(l+3)·O(s·{tilde over (m)}·ñ·{tilde over (q)})+(l+5)·O({tilde over (m)}·ñ),  Eq. 38

where l is the number of trials in step 278 in the search for β^((k)). By assuming that Alg. 3 converges within k₁ iterations, we further approximate the cost in Eq. 38 as,

COST₁ ≈k ₁·(l+3)·O(s·{tilde over (m)}·ñ·{tilde over (q)})+(l+5)·O({tilde over (m)}·ñ).  Eq. 39

The most expensive part stems from the first term in Eq. 39, which is dominated by step 278 and step 282 in method 270, and the number of iterations in Alg. 3. The cost of Alg. 4 is more straightforward. It includes the cost from the Gauss-Seidel iteration in Eq. 35 (step 308), and the shrinkage formula in Eqs. 36 and 37 (step 310). Hence we have Eq. 40:

COST₂≈18·O({tilde over (m)}·ñ).  Eq. 40

The total computational cost of Alg. 2 becomes,

COST≈k ₂ ·[k ₁·(l+3)·O(s·{tilde over (m)}·ñ·{tilde over (q)})+(l+23)·O({tilde over (m)}·ñ)],  Eq. 41

where k₂ is the total number of iterations in Alg. 3. We see from Eq. 41 that the extra cost of solving Eq. 25 is trivial compared to the cost of solving Eq. 24.

Two groups of tests were performed to illustrate the feasibility and efficiency of the ultrasound waveform tomography method with modified total-variation regularization in accordance with the present invention. Ultrasound waveform tomography reconstructions were first provided using the Tikhonov regularization and the conventional TV regularization for comparison. Results for numerical breast phantoms with tumors of different diameters are shown to further illustrate the improved resolution of the ultrasound waveform tomography method of the present invention compared to that of the other methods.

We used synthetic-aperture ultrasound data from two parallel transducer arrays to test the capability of ultrasound waveform tomography with the modified TV regularization scheme of the present invention for reconstructing the sound speed of small breast tumors. We also compare the result with that obtained using the Tikhonov regularization and the conventional TV regularization. The geometry of synthetic-aperture ultrasound tomography system with two parallel transducer arrays is schematically illustrated in FIG. 5. Each transducer array 14 a and 14 b comprises of 384 transducer elements 16 evenly distributed, with a pitch size of 0.55 mm. The two transducer arrays 14 a and 14 b are separated by 20 cm. The ultrasound source function used is a Ricker wavelet with a central frequency of 1.0 MHz.

A numerical breast phantom as shown in FIG. 27A to test the ultrasound waveform tomography methods. The background sound speed is 1500 m/s. The sound speeds of the two small tumors are 1545 m/s and 1560 ms and their diameters are 2.2 mm and 3.0 mm, respectively. These diameters represent approximately 1.5 wavelengths and 2.0 wavelengths.

We conducted ultrasound waveform tomography with the Tikhonov regularization using simulated ultrasound transmission and reflection data, and produce the reconstruction image in FIG. 28A. Both tumors are reasonably well reconstructed. However, the reconstructed sound-speed value of the smaller tumor is smaller than its true value. Moreover, the interfaces of the two objects are smoothed to some extent.

We applied ultrasound waveform tomography with the conventional TV regularization scheme to the same synthetic-aperture ultrasound data, and produced the reconstruction image in FIG. 28B. The blurring effect of the Tikhonov reconstruction has been mostly eliminated. The edges of the two small breast tumors are very well preserved. However, we also notice that there are some “ringing” artifacts generated in the inner regions of the tumors.

We then employed the ultrasound waveform tomography with the modified TV regularization to the same data, and show the reconstruction result in FIG. 28C. By comparing to FIG. 28A and FIG. 28B, we see that not only the edges have been very well preserved, but also the “ringing” artifacts have been mostly eliminated.

To better visualize the quantitative sound-speed values, we plotted the horizontal profiles for FIG. 28A through FIG. 28C at two different locations: one at the vertical position of 103 mm as shown in FIG. 30A through FIG. 30C, and the other at the vertical position of 98 mm as displayed in FIG. 29A through FIG. 29C. For the smaller breast tumor, the waveform tomography reconstruction using the Tikhonov regularization gives the sound speed value of 1540 m/s, while the waveform tomography reconstruction using the TV or modified TV regularization schemes yields better approximated values to its true value of 1545 m/s and well preserves the edges of the tumors. However, the waveform tomography reconstruction result with the conventional TV regularization is much more oscillatory than the result obtained with ultrasound waveform tomography with the modified TV regularization. Therefore, the waveform tomography reconstruction with the modified TV regularization produces an even more accurate value of the sound speed than that using the conventional TV regularization.

In order to study the resolution of our new ultrasound waveform tomography method for tumors with small sizes, we created four numerical breast phantoms with different diameters ranging from 0.75 mm (about a half of a wavelength) to 4.5 mm (about 3 wavelengths). The tumors diameters in millimeters are 0.75 and 1.5; 1.5 and 2.25; 3.0 and 3.75; 3.75 and 4.5. The sound speeds of the background medium, small tumor and large tumor, are set to be 1500 m/s, 1540 m/s and 1560 m/s.

Besides the 2D reconstruction results, we also show the two horizontal profiles across the centers of the two tumors. As illustrated in FIG. 31A through FIG. 31C and FIG. 32A through FIG. 32D, we observe that the ultrasound waveform tomography method with the modified TV regularization can characterize tumors as small as about a half of an ultrasound wavelength. As we gradually increase the sizes of the tumors, as in FIG. 33A through FIG. 33C and FIG. 34A through FIG. 34D, FIG. 35A through FIG. 35C and FIG. 36A through FIG. 36D, and FIG. 37A through FIG. 37C and FIG. 38A through FIG. 38D, our method consistently provides accurate reconstructions of sound speeds and the shapes of the tumors.

In summary, the synthetic-aperture ultrasound tomography systems and methods of the present invention acquire ultrasound transmission and reflection data at the same time, and we have demonstrated that ultrasound waveform tomography using both ultrasound transmission and reflection data simultaneously greatly improves tomographic reconstructions of shapes and sound-speeds of tumors compared to tomographic reconstructions using only transmission data or only reflection data, particularly when used with the modified TV regularization method of the present invention.

From the discussion above it will be appreciated that the invention can be embodied in various ways, including the following:

1. A synthetic aperture ultrasound tomography imaging method for imaging a tissue medium with one or more ultrasound transducer arrays comprising a plurality of transducers, the method comprising: exciting a first transducer with plurality of transducers to generate an ultrasound field within the tissue medium; acquiring a transmission signal and a reflection signal from a second transducer within the one or more ultrasound transducer arrays; and generating an ultrasound waveform tomography image reconstruction using both the acquired reflection and transmission signals.

2. A method as recited in any of the preceding embodiments, wherein said step of generating an ultrasound waveform tomography image reconstruction is a function of computing an acoustic wave property of the reflection and transmission signals by calculating a minimum mean square difference between observed and synthetic waveforms relating to the reflection and transmission signals.

3. A method as recited in any of the preceding embodiments, wherein said image reconstruction is a function of:

${{E(m)} = {\min\limits_{m}\left\{ {{d - {f(m)}}}_{2}^{2} \right\}}},$

where ∥d−ƒ(m)∥₂ ² comprises a misfit function, and d comprises data relating to the acquired reflection signal and transmission signals.

4. A method as recited in any of the preceding embodiments, further comprising performing total-variation regularization to generate sound-speed reconstructions of the acquired reflection and transmission signals.

5. A method as recited in any of the preceding embodiments 4, wherein said total-variation regularization comprises a misfit function.

6. A method as recited in any of the preceding embodiments, further comprising obtaining a gradient of the misfit function using an adjoint state method.

7. A method as recited in any of the preceding embodiments, wherein said total-variation regularization is a function of:

${{E(m)} = {\min\limits_{m}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda {m}_{TV}}} \right\}}},$

and where ∥d−ƒ(m)∥₂ ² comprises a misfit function, d comprises data relating to the acquired reflection signal and transmission signals, λ, is a positive regularization parameter, and ∥m∥_(TV) is a TV regularization term.

8. A method as recited in any of the preceding embodiments, wherein λ, is selected as a function of:

${\lambda = \frac{{{d - {f(m)}}}_{2}^{2}}{r \cdot {m}_{TV}}},$

where ∥d−f(m)∥₂ ² is a fidelity term, ∥m∥_(TV) is a regularization term and a value of r is defined to be the ratio between the data fidelity term and the regularization term.

9. A method as recited in any of the preceding embodiments, wherein said total-variation regularization comprises a modified total-variation regularization that is a function of:

${{E\left( {m,u} \right)} = {\min\limits_{m,u}\left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda_{1}{{m - u}}_{2}^{2}} + {\lambda_{2}{{\nabla u}}_{1}}} \right\}}},$

and where ∥d−ƒ(m)∥₂ ² comprises a data misfit function, d comprises data relating to the acquired reflection signal and transmission signal, where ∥₁ and λ₂ are both positive regularization parameters, and u is an auxiliary variable.

10. A method as recited in any of the preceding embodiments, wherein λ₁ and λ₂ are selected as a function of:

${\lambda_{i} = \frac{{data}\mspace{14mu} {fidelity}\mspace{14mu} {term}}{{r_{i} \cdot {regularization}}\mspace{14mu} {term}}},{i = {1,2,}}$

where the values of r_(i) are defined to be the ratio between the data fidelity term and the regularization term.

11. A method as recited in any of the preceding embodiments: wherein said total-variation regularization comprises a cost function; and wherein said cost function is decomposed with a modified TV regularization into two regularization problems.

12. A method as recited in any of the preceding embodiments, wherein said regularization problems comprise a L₂-norm-based Tikhonov regularization problem and a L₁-norm-based TV regularization problem.

13. A method as recited in any of the preceding embodiments, wherein the misfit function is minimized using an alternating minimization algorithm.

14. A method as recited in any of the preceding embodiments, wherein a nonlinear conjugate gradient (NCG) approach is used to solve for the Tikhonov regularization problem.

15. A method as recited in any of the preceding embodiments, wherein a split-Bregman method is used to solve the L₁-norm-based TV regularization problem.

16. A method as recited in any of the preceding embodiments: wherein the plurality of transducers are configured such that a first set of two or more transducers are positioned at an opposing spaced-apart orientation from a second set of two or more transducers such that the first set of two or more transducers face the second set of two or more transducers; wherein the first and second sets of two or more transducers are positioned at spaced-apart locations so as to allow for the tissue medium to be positioned in between the first and second sets of two or more transducers; and wherein the method further comprises: exciting a first transducer with the first set of two or more transducers to generate an ultrasound field within the tissue medium; and receiving a transmission signal and a reflection signal from at least the second set of two or more transducers.

17. A method as recited in any of the preceding embodiments, further comprising receiving a reflection signal from all transducers in the one or more arrays.

18. A method as recited in any of the preceding embodiments, further comprising simultaneously receiving the reflection and transmission signals from the second set of two or more transducers.

19. A synthetic aperture ultrasound tomography imaging system for imaging a tissue medium with one or more ultrasound transducer arrays comprising a plurality of transducers, the system comprising: a processor; and programming executable on said processor and configured for: exciting a first transducer with plurality of transducers to generate an ultrasound field within the tissue medium; receiving a transmission signal and a reflection signal from a second transducer within the one or more ultrasound transducer arrays; and generating an ultrasound waveform tomography image reconstruction using both the acquired reflection and transmission signals.

20. A system as recited in any of the preceding embodiments, wherein said step of generating an ultrasound waveform tomography image reconstruction is a function of computing an acoustic wave property of the reflection and transmission signals by calculating a minimum mean square difference between observed and synthetic waveforms relating to the reflection and transmission signals.

21. A system as recited in any of the preceding embodiments, wherein the image reconstruction is a function of:

${{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{d - {f(m)}}}_{2}^{2} \right\}}},$

where ∥d−ƒ(m)∥₂ ² comprises a misfit function, and d comprises data relating to the acquired reflection signal and transmission signal.

22. A system as recited in any of the preceding embodiments, wherein total-variation regularization is performed to generate sound-speed reconstructions of the acquired reflection and transmission signals.

23. A system as recited in any of the preceding embodiments, wherein said total-variation regularization comprises a misfit function.

24. A system as recited in any of the preceding embodiments, wherein a gradient of the misfit function is obtained using an adjoint state system.

25. A system as recited in any of the preceding embodiments, wherein said total-variation regularization is a function of:

${{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda {m}_{TV}}} \right\}}},$

and where ∥d−ƒ(m)∥₂ ² comprises a misfit function, d comprises data relating to the acquired reflection signal and transmission signals, λ, is a positive regularization parameter, and ∥m∥_(TV) is a TV regularization term.

26. A system as recited in any of the preceding embodiments 22, wherein λ is selected as a function of:

${\lambda = \frac{{{d - {f(m)}}}_{2}^{2}}{r \cdot {m}_{TV}}},$

where ∥d−f(m)∥₂ ² is a fidelity term, ∥m∥_(TV) is a regularization term and a value of r is defined to be the ratio between the data fidelity term and the regularization term.

27. A system as recited in any of the preceding embodiments, wherein said total-variation regularization comprises a modified total-variation regularization that is a function of:

${{E\left( {m,u} \right)} = {\min\limits_{m,u}\mspace{14mu} \left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda_{1}{{m - u}}_{2}^{2}} + {\lambda_{2}{\nabla_{u}}_{1}}} \right\}}},$

and where ∥d−ƒ(m)∥₂ ² comprises a data misfit function, d comprises data relating to the acquired reflection signal and transmission signal, where λ₁ and λ₂ are both positive regularization parameters, and u is an auxiliary variable.

28. A system as recited in any of the preceding embodiments, wherein λ₁ and λ₂ are selected as a function of:

${\lambda_{i} = \frac{{data}\mspace{14mu} {fidelity}\mspace{14mu} {term}}{{r_{i} \cdot {regularization}}\mspace{14mu} {term}}},{i = 1},2,$

where the values of r_(i) are defined to be the ratio between the data fidelity term and the regularization term.

29. A system as recited in any of the preceding embodiments: wherein said total-variation regularization comprises a cost function; and wherein said cost function is decomposed with a modified TV regularization into two regularization problems.

30. A system as recited in any of the preceding embodiments, wherein said regularization problems comprise a L₂-norm-based Tikhonov regularization problem and a L₁-norm-based TV regularization problem.

31. A system as recited in any of the preceding embodiments, wherein the misfit function is minimized using an alternating minimization algorithm.

32. A system as recited in any of the preceding embodiments, wherein a nonlinear conjugate gradient (NCG) approach is used to solve for the Tikhonov regularization problem.

33. A system as recited in any of the preceding embodiments, wherein a split-Bregman system to is used to solve the L₁-norm-based TV regularization problem.

34. A synthetic system as recited in any of the preceding embodiments: wherein the plurality of transducers are configured such that a first set of two or more transducers are positioned at an opposing spaced-apart orientation from a second set of two or more transducers such that the first set of two or more transducers face the second set of two or more transducers; wherein the first and second sets of two or more transducers are positioned at spaced-apart locations so as to allow for the tissue medium to be positioned in between the first and second sets of two or more transducers; and wherein the programming is further configured for: exciting a first transducer with the first set of two or more transducers to generate an ultrasound field within the tissue medium; and receiving a transmission signal and a reflection signal from at least the second set of two or more transducers.

35. A system as recited in any of the preceding embodiments 34, further comprising receiving a reflection signal from all transducers in the one or more arrays.

36. A system as recited in any of the preceding embodiments, wherein the programming is further configured for simultaneously receiving the reflection and transmission signals from the second set of two or more transducers.

37. A synthetic aperture ultrasound tomography imaging system for imaging a tissue medium, the system comprising: one or more ultrasound transducer arrays; said one or more ultrasound transducer arrays comprising a plurality of transducers; a processor; and programming executable on said processor and configured for: exciting a first transducer with plurality of transducers to generate an ultrasound field within the tissue medium; receiving a transmission signal and a reflection signal from a second transducer within the one or more ultrasound transducer arrays; and generating an ultrasound waveform tomography image reconstruction using both the acquired reflection and transmission signals.

38. A system as recited in any of the preceding embodiments 37: wherein the plurality of transducers are configured such that a first set of two or more transducers are positioned at an opposing spaced-apart orientation from a second set of two or more transducers such that the first set of two or more transducers face the second set of two or more transducers; and wherein the first and second sets of two or more transducers are positioned at spaced-apart locations so as to allow for the tissue medium to be positioned in between the first and second sets of two or more transducers.

Embodiments of the present invention may be described with reference to flowchart illustrations of methods and systems according to embodiments of the invention, and/or algorithms, formulae, or other computational depictions, which may also be implemented as computer program products. In this regard, each block or step of a flowchart, and combinations of blocks (and/or steps) in a flowchart, algorithm, formula, or computational depiction can be implemented by various means, such as hardware, firmware, and/or software including one or more computer program instructions embodied in computer-readable program code logic.

As will be appreciated, any such computer program instructions may be loaded onto a computer, including without limitation a general purpose computer or special purpose computer, or other programmable processing apparatus to produce a machine, such that the computer program instructions which execute on the computer or other programmable processing apparatus create means for implementing the functions specified in the block(s) of the flowchart(s).

Accordingly, blocks of the flowcharts, algorithms, formulae, or computational depictions support combinations of means for performing the specified functions, combinations of steps for performing the specified functions, and computer program instructions, such as embodied in computer-readable program code logic means, for performing the specified functions. It will also be understood that each block of the flowchart illustrations, algorithms, formulae, or computational depictions and combinations thereof described herein, can be implemented by special purpose hardware-based computer systems which perform the specified functions or steps, or combinations of special purpose hardware and computer-readable program code logic means.

Furthermore, these computer program instructions, such as embodied in computer-readable program code logic, may also be stored in a computer-readable memory that can direct a computer or other programmable processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the block(s) of the flowchart(s). The computer program instructions may also be loaded onto a computer or other programmable processing apparatus to cause a series of operational steps to be performed on the computer or other programmable processing apparatus to produce a computer-implemented process such that the instructions which execute on the computer or other programmable processing apparatus provide steps for implementing the functions specified in the block(s) of the flowchart(s), algorithm(s), formula(e), or computational depiction(s).

Although the description herein contains many details, these should not be construed as limiting the scope of the disclosure but as merely providing illustrations of some of the presently preferred embodiments. Therefore, it will be appreciated that the scope of the disclosure fully encompasses other embodiments which may become obvious to those skilled in the art.

In the claims, reference to an element in the singular is not intended to mean “one and only one” unless explicitly so stated, but rather “one or more.” All structural, chemical, and functional equivalents to the elements of the disclosed embodiments that are known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the present claims. Furthermore, no element, component, or method step in the present disclosure is intended to be dedicated to the public regardless of whether the element, component, or method step is explicitly recited in the claims. No claim element herein is to be construed as a “means plus function” element unless the element is expressly recited using the phrase “means for”. No claim element herein is to be construed as a “step plus function” element unless the element is expressly recited using the phrase “step for”. 

1. A synthetic aperture ultrasound tomography imaging method for imaging a tissue medium with one or more ultrasound transducer arrays comprising a plurality of transducers, the method comprising: exciting a first transducer with plurality of transducers to generate an ultrasound field within the tissue medium; acquiring a transmission signal and a reflection signal from a second transducer within the one or more ultrasound transducer arrays; and generating an ultrasound waveform tomography image reconstruction using both the acquired reflection and transmission signals.
 2. A method as recited in claim 1, wherein said step of generating an ultrasound waveform tomography image reconstruction is a function of computing an acoustic wave property of the reflection and transmission signals by calculating a minimum mean square difference between observed and synthetic waveforms relating to the reflection and transmission signals.
 3. A method as recited in claim 2, wherein said image reconstruction is a function of: ${{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{d - {f(m)}}}_{2}^{2} \right\}}},$ where ∥d−ƒ(m)∥₂ ² comprises a misfit function, and d comprises data relating to the acquired reflection signal and transmission signals.
 4. A method as recited in claim 1, further comprising performing total-variation regularization to generate sound-speed reconstructions of the acquired reflection and transmission signals.
 5. A method as recited in claim 4, wherein said total-variation regularization comprises a misfit function.
 6. A method as recited in claim 5, further comprising obtaining a gradient of the misfit function using an adjoint state method.
 7. A method as recited in claim 4, wherein said total-variation regularization is a function of: ${{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda {m}_{TV}}} \right\}}},$ and where ∥d−ƒ(m)∥₂ ² comprises a misfit function, d comprises data relating to the acquired reflection signal and transmission signals, λ is a positive regularization parameter, and ∥m∥_(TV) is a TV regularization term.
 8. A method as recited in claim 4, wherein λ is selected as a function of: ${\lambda = \frac{{{d - {f(m)}}}_{2}^{2}}{r \cdot {m}_{TV}}},$ where ∥d−f(m)∥₂ ² is a fidelity term, ∥m∥_(TV) is a regularization term and a value of r is defined to be the ratio between the data fidelity term and the regularization term.
 9. A method as recited in claim 4, wherein said total-variation regularization comprises a modified total-variation regularization that is a function of: ${{E\left( {m,u} \right)} = {\min\limits_{m,u}\mspace{14mu} \left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda_{1}{{m - u}}_{2}^{2}} + {\lambda_{2}{\nabla_{u}}_{1}}} \right\}}},$ and where ∥d−ƒ(m)∥₂ ² comprises a data misfit function, d comprises data relating to the acquired reflection signal and transmission signal, where λ₁ and λ₂ are both positive regularization parameters, and u is an auxiliary variable.
 10. A method as recited in claim 4, wherein λ₁ and λ₂ are selected as a function of: ${\lambda_{i} = \frac{{data}\mspace{14mu} {fidelity}\mspace{14mu} {term}}{{r_{i} \cdot {regularization}}\mspace{14mu} {term}}},{i = 1},2,$ where the values of r_(i) are defined to be the ratio between the data fidelity term and the regularization term. 11-18. (canceled)
 19. A synthetic aperture ultrasound tomography imaging system for imaging a tissue medium with one or more ultrasound transducer arrays comprising a plurality of transducers, the system comprising: a processor; and programming executable on said processor and configured for: exciting a first transducer with plurality of transducers to generate an ultrasound field within the tissue medium; receiving a transmission signal and a reflection signal from a second transducer within the one or more ultrasound transducer arrays; and generating an ultrasound waveform tomography image reconstruction using both the acquired reflection and transmission signals.
 20. A system as recited in claim 19, wherein said step of generating an ultrasound waveform tomography image reconstruction is a function of computing an acoustic wave property of the reflection and transmission signals by calculating a minimum mean square difference between observed and synthetic waveforms relating to the reflection and transmission signals.
 21. A system as recited in claim 20, wherein the image reconstruction is a function of: ${{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{d - {f(m)}}}_{2}^{2} \right\}}},$ where ∥d−ƒ(m)∥₂ ² comprises a misfit function, and d comprises data relating to the acquired reflection signal and transmission signal.
 22. A system as recited in claim 19, wherein total-variation regularization is performed to generate sound-speed reconstructions of the acquired reflection and transmission signals.
 23. A system as recited in claim 22, wherein said total-variation regularization comprises a misfit function.
 24. A system as recited in claim 23, wherein a gradient of the misfit function is obtained using an adjoint state system.
 25. A system as recited in claim 22, wherein said total-variation regularization is a function of: ${{E(m)} = {\min\limits_{m}\mspace{14mu} \left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda {m}_{TV}}} \right\}}},$ and where ∥d−ƒ(m)∥₂ ² comprises a misfit function, d comprises data relating to the acquired reflection signal and transmission signals, λ is a positive regularization parameter, and ∥m∥_(TV) is a TV regularization term.
 26. A system as recited in claim 22, wherein λ is selected as a function of: ${\lambda = \frac{{{d - {f(m)}}}_{2}^{2}}{r \cdot {m}_{TV}}},$ where ∥d−f(m)∥₂ ² is a fidelity term, ∥m∥_(TV) is a regularization term and a value of r is defined to be the ratio between the data fidelity term and the regularization term.
 27. A system as recited in claim 22, wherein said total-variation regularization comprises a modified total-variation regularization that is a function of: ${{E\left( {m,u} \right)} = {\min\limits_{m,u}\mspace{14mu} \left\{ {{{d - {f(m)}}}_{2}^{2} + {\lambda_{1}{{m - u}}_{2}^{2}} + {\lambda_{2}{\nabla_{u}}_{1}}} \right\}}},$ and where ∥d−ƒ(m)∥₂ ² comprises a data misfit function, d comprises data relating to the acquired reflection signal and transmission signal, where λ₁ and λ₂ are both positive regularization parameters, and u is an auxiliary variable.
 28. A system as recited in claim 22, wherein λ₁ and λ₂ are selected as a function of: ${\lambda_{i} = \frac{{data}\mspace{14mu} {fidelity}\mspace{14mu} {term}}{{r_{i} \cdot {regularization}}\mspace{14mu} {term}}},{i = 1},2,$ where the values of r_(i) are defined to be the ratio between the data fidelity term and the regularization term. 29-38. (canceled) 